Find the circumstance of the triangle whose vertices are given by the complex numbers `z_(1),z_(2)` and `z_(3)`.

Show that the area of the triangle on the Argand diagram formed by the complex numbers z, zi and z+ zi is =(1)/(2) |z|^(2)

Find the co-ordinates of the centroid of the triangle whose vertices are (x_(1),y_(1),z_(1)), (x_(2),y_(1),z_(2)) and (x_(3),y_(3),z_(3)) .

The area of the triangle with vertices affixed at z, iz, z(1 +i) is:

Let A, B, C, D and E be points on the complex plane, which respectively represent the complex numbers z_1,z_2,z_3, z_4 and z_5 . If (z_3 - z_2)z_4 =(z_1- z_2)z_5 , prove that the triangles ABC and DOE are similar.

Find the difference of the complex numbers : z_1=- 3 + 2i and z_2= 13-i .

Let z = x + iy be a complex number where x and y are integers. Then ther area of the rectangle whose vertices are the roots of the equaiton barz z^(3) + zbarz^(3) = 350 .

If |z|=2, the points representing the complex numbers -1+5z will lie on

A,B and C are the points respectively the complex numbers z_(1),z_(2) and z_(3) respectivley, on the complex plane and the circumcentre of /_\ABC lies at the origin. If the altitude of the triangle through the vertex. A meets the circumcircle again at P, prove that P represents the complex number (-(z_(2)z_(3))/(z_(1))) .

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

If z, izand z+iz are the vertices of a triangle whose area is 2units, the value of |z| is